Adding image texture to procedural texture without colour alteration
You are using a VectorMath node there, rather than a color blending node.
As Craig D Jones said, MixRGB is correct for color blending (like 'photoshop-layer' blending rules).

It's unclear what your goal is, but you appear to mean you want to just 'alpha over' the two inputs, when you say 'without affecting each other'.

One way to achieve that would be to pass the color data to Diffuse BSDFs, then use a Mix Shader to combine these (with 0.5 giving the middle result of some from each), and then take that shader output into a 'Converter > Shader to RGB' Node.

The 'Shader to RGB' node is under 'Converter', 3 items from the bottom of the menu, assuming you are editing an Object's Material nodes. It is not available in World nodes.